Problem 21 - Prealgebra

Question

Solve each of these problems using the method developed in this section. When manufacturing a product, a certain amount of variation (or error) can occur in the process and still create a part or product that is useable. For one particular company, a \(3 \%\) error is acceptable for their machine parts to be used safely. If the part they are manufacturing is 22.5 in. long, what is the range of measures that are acceptable for this part?

Step-by-Step Solution

Answer

The acceptable range is from 21.825 inches to 23.175 inches.

Step 1: Understand the problem

We need to find the acceptable length range for a part that is supposed to be 22.5 inches long. A 3% error is acceptable, meaning the actual length can vary within ±3% of the target length.

Step 2: Calculate the magnitude of acceptable error

Calculate 3% of the target length (22.5 inches) to find the amount by which the length can deviate. This is calculated as \( 0.03 \times 22.5 \).

Step 3: Perform the calculation

Calculate the amount of acceptable error: \( 0.03 \times 22.5 = 0.675 \). This means the part can be 0.675 inches longer or shorter than 22.5 inches.

Step 4: Find the acceptable range of measures

Apply the error to the target length to find the range. The lower bound is \( 22.5 - 0.675 \) and the upper bound is \( 22.5 + 0.675 \).

Step 5: Calculate the range limits

Calculate the lower limit: \( 22.5 - 0.675 = 21.825 \) inches. Calculate the upper limit: \( 22.5 + 0.675 = 23.175 \) inches.

Step 6: Express the final range

The acceptable range of measures for the part is from 21.825 inches to 23.175 inches.

Key Concepts

Percentage Error

When working with measurements, understanding percentage error helps us determine how much a value may deviate from its target. Let's say we're aiming for a measurement of 22.5 inches, like in our example. A percentage error is simply the proportion of this value that we're willing to accept as variation, expressed as a percentage.
For instance, in our exercise, a 3% error is acceptable. To find what 3% of 22.5 inches means, perform the calculation:

• First, convert the percentage to a decimal by dividing by 100. So, 3% becomes 0.03.
• Multiply this decimal by the original measurement: \( 0.03 \times 22.5 \).
• This calculation gives us the actual error amount, which is 0.675 inches.

This means the part can be somewhat shorter or longer than 22.5 inches, but not more than 0.675 inches off the target.

Range of Values

Once the percentage error is determined, we can find the range of acceptable values for our measurement. This range tells us the smallest and largest acceptable measurements.Here's how you establish this range:

• Take the target measurement, which in this scenario is 22.5 inches.
• Subtract the error amount from the target for the lower end of the range: \( 22.5 - 0.675 = 21.825 \) inches.
• Add the error amount to the target for the upper end: \( 22.5 + 0.675 = 23.175 \) inches.

Thus, any length within 21.825 inches and 23.175 inches is considered acceptable. This entire spread is our range of values, ensuring parts fit and function correctly while acknowledging slight variations in manufacturing.

Acceptable Limits

In manufacturing and engineering, setting acceptable limits ensures the products work as expected while allowing for minor errors. Acceptable limits are the boundaries within which your measurements must fall to be deemed usable. They serve as a guide for quality control, helping manufacturers identify when a product is outside of safe specifications.
For our machine part:

• The part should measure between 21.825 and 23.175 inches, based on a 3% allowed variation.
• Observing these limits ensures that every part is safe, functional, and up to standard.

In practice, regularly measuring and testing parts against these limits allows manufacturers to maintain quality and safety, reducing the risk of failure or malfunctions.